U.S. patent application number 14/410019 was filed with the patent office on 2015-07-09 for method for determining the size distribution of a mixture of particles using taylor dispersion, and associated system.
The applicant listed for this patent is CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIOUE (C.N.R.S), UNIVERSITE MONTPELLIER 2, SCIENCES ET TECHBIQUES, UNIVERSITE MONTPELLIER I. Invention is credited to Jean-Philippe Biron, Luca Cipelletti, Herve Cottet, Michel Martin.
Application Number | 20150192507 14/410019 |
Document ID | / |
Family ID | 47049253 |
Filed Date | 2015-07-09 |
United States Patent
Application |
20150192507 |
Kind Code |
A1 |
Cottet; Herve ; et
al. |
July 9, 2015 |
METHOD FOR DETERMINING THE SIZE DISTRIBUTION OF A MIXTURE OF
PARTICLES USING TAYLOR DISPERSION, AND ASSOCIATED SYSTEM
Abstract
The method comprises the following steps: injecting (100) a
sample into a capillary; transporting (110) the sample along the
capillary in experimental conditions suited to generate a Taylor
dispersion phenomenon; generating (120) a signal characteristic of
the Taylor dispersion; processing (130) the signal in order to
obtain the experimental Taylor signal S(t); and analysing (200) the
experimental Taylor signal S(t), The analysis step consists of
seeking an amplitude distribution P(G.sub.(c)) that allows the
experimental Taylor signal S(t) to be broken down into a sum of
Gaussian functions by implementing a constrained regularization
algorithm consisting of minimising a cost function H.sub..alpha.
including at least one constraint term associated with a constraint
that must observe the amplitude distribution P(G.sub.(c)) , whereby
the minimization is carried out on an interval of interest of the
values of the parameter G.sub.(c) that is characteristic of the
Gaussian amplitude function P(G.sub.(c)).
Inventors: |
Cottet; Herve; (Le Cres,
FR) ; Cipelletti; Luca; (Montpellier, FR) ;
Martin; Michel; (Le Plessis-Pate, FR) ; Biron;
Jean-Philippe; (Saint Gely Du Fesc, FR) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIOUE (C.N.R.S)
UNIVERSITE MONTPELLIER I
UNIVERSITE MONTPELLIER 2, SCIENCES ET TECHBIQUES |
Paris
Montpellier Cedex 1
Montepellier |
|
FR
FR
FR |
|
|
Family ID: |
47049253 |
Appl. No.: |
14/410019 |
Filed: |
June 26, 2013 |
PCT Filed: |
June 26, 2013 |
PCT NO: |
PCT/EP2013/063432 |
371 Date: |
December 19, 2014 |
Current U.S.
Class: |
702/29 |
Current CPC
Class: |
G01N 15/0205 20130101;
G01N 15/0211 20130101; G01N 35/085 20130101; G01N 15/02
20130101 |
International
Class: |
G01N 15/02 20060101
G01N015/02 |
Foreign Application Data
Date |
Code |
Application Number |
Jun 26, 2012 |
FR |
1256050 |
Claims
1. Method for determining the size distribution of a mixture of
molecule or particle species comprising the following steps:
injecting a sample of the mixture to be analyzed inside a capillary
in which an eluent is flowing; transporting the sample injected
along the capillary from an injection section to a detection
section thereof, in experimental conditions suitable to generate a
Taylor dispersion phenomenon that is measurable at the level of the
detection section; generating, by means of a suitable sensor
included in the detection section, a signal characteristic of the
Taylor dispersion of the transported sample; processing the
detection signal in order to obtain an experimental Taylor signal
S(t); and analyzing the experimental Taylor signal S(t), wherein
the step of analyzing an experimental Taylor signal S(t) of a
sample of the mixture consists of searching an amplitude
distribution P(G.sub.(c)) that allows the experimental Taylor
signal S(t) to be broken down into a sum of Gaussian functions by
means of the equation I: {circumflex over
(S)}(t).ident..intg..sub.0.sup..infin.P(G.sub.(c))G.sub.(c).sup.c/2exp[-(-
t-t.sub.0).sup.2G.sub.(c).sup.c]dG.sub.(c) (I) where t is a
variable upon which the experimental Taylor signal depends and
t.sub.0 is a value of the variable t common to the various Gaussian
functions and corresponding to the peak of the experimental Taylor
signal S(t); G.sub.(c) is a characteristic parameter of a Gaussian
amplitude function P(G.sub.(c)) and is associated: where c=1, with
the diffusion coefficient D of a species according to the relation
G.sub.(1)=12D/(R.sub.c.sup.2t.sub.0) for c=-1, to the hydrodynamic
ray R.sub.h of a species according to the relation G ( - 1 ) = 2 k
B T .pi. .eta. R c 2 t 0 R h - 1 ; ##EQU00047## and for
c=-1/d.sub.f=-(1+a)/3, to the molar mass M of a species according
to the relation G = 2 k B T .pi. .eta. R c 2 t 0 ( 10 .pi. N a 3 K
) 1 / 3 M - ( 1 + a 3 ) , ##EQU00048## where k.sub.B is the
Boltzmann constant, T is the absolute temperature expressed in
Kelvins at which the experiment is conducted, .eta. is the
viscosity of the eluent used, R.sub.c is the internal ray of the
capillary used, Na.sub..alpha. is Avogadro's number, and K and a
are Mark Houwink coefficients, by implementing a constrained
regularization algorithm consisting of minimizing a cost function
H.sub..alpha. including at least one constraint term associated
with a constraint that must observe the amplitude distribution
P(G.sub.(c)) that is the solution of the foregoing equation,
whereby the minimization is carried out on an interval of interest
of the values of the parameter G.sub.(c).
2. Method according to claim 1, wherein equation I is discretized
by subdividing the interval of values of the parameter G.sub.(c),
whereby each discretization point G.sub.m is indexed by an integer
m that varies between the unit value and the value N, whereby the
point G.sub.m is at a distance from the point G.sub.m-1
corresponding to a sub-interval of the length c.sub.m.
3. Method according to claim 2, wherein the cost function takes the
following form:
H.sub..alpha.=.chi..sup.2+.alpha..sup.2.DELTA..sup.2 where: the
first term .chi..sup.2 is a distance term between the experimental
Taylor signal S(t) and a reconstructed Taylor signal defined by:
S'(t)=.SIGMA..sub.m=1.sup.Nc.sub.mP(G.sub.m) {square root over
(G.sub.m)}exp[-(t-t.sub.0).sup.2G.sub.m], and the second term
.DELTA..sup.2 is a constraint term associated with the at least one
constraint that must observe the amplitude distribution P(G) that
is the solution of the foregoing equation, whereby the second term
is introduced by a Lagrange coefficient .alpha., allowing the
contribution of the second term of the cost function H.sub..alpha.
to be adapted to the first term.
4. Method according to claim 3, wherein the first term .chi..sup.2
is a distance of the type `least squares` taking the form:
.chi..sup.2=.SIGMA..sub.k=1.sup.L(S'(t.sub.k)-{circumflex over
(S)}'(t.sub.k)).sup.2 where the experimental Taylor signal S(t) and
the reconstructed function S(t) are sampled over time, whereby each
sample is indexed by an integer k varying between the unit value
and the value L.
5. Method according to claim 3, wherein the at least one constraint
that must be observed by the amplitude distribution P(G) that is
the solution of the foregoing equation is a regularity constraint
associated with a constraint term .DELTA..sup.2 preferably taking
the form:
.DELTA..sup.2=.SIGMA..sub.m=2.sup.N-1[P(G.sub.m-1)-2P(G.sub.m)+P(G.sub.m+-
1)].sup.2.
6. Method according to claim 3, wherein the analysis step includes
a step of determining the optimal value .alpha..sub.0 of the
Lagrange coefficient .alpha. such that the value of the distance
term .chi..sup.2 corresponding to the minimum of the cost function
H.sub..alpha.=.alpha.0 is close by values lower than a statistical
error .nu., preferably of the form .nu.L-N.
7. Method according to claim 3, wherein, because the interval of
interest of the values of the parameter G.sub.(c) are delimited by
a minimum G.sub.min and a maximum G.sub.max, the method includes a
step of determining the values of the minimum and maximum.
8. Method according to claim 7, comprising a step of breaking down
a normalized Taylor signal s(t) associated with the experimental
Taylor signal S(t) into components by the relation:
s(t)=S(t)/S(t.sub.0), consisting of adjusting to the curve ln[s(t)]
a second-order polynomial of the variable (t-t.sub.0).sup.2 so as
to determine the first-order components .GAMMA..sub.1 and
second-order components .GAMMA..sub.2, and in that the step of
determining the values of the minimum G.sub.min and maximum
G.sub.max uses the equations: .beta. = ln .GAMMA. 1 - ln ( 1 +
.GAMMA. 2 .GAMMA. 1 2 ) ##EQU00049## and ##EQU00049.2## .gamma. =
ln ( 1 + .GAMMA. 2 .GAMMA. 1 2 ) ##EQU00049.3## where .beta. and
.gamma. are respectively the average and the standard deviation of
the logarithm of the parameter G of a log-normal distribution,
followed by, G.sub.min=exp(.beta.-k {square root over (2)}.gamma.)
and G.sub.max=exp(.beta.+k {square root over (2)}.gamma.).
9. Method according to claim 7, wherein the step of determining the
values of the minimum and maximum of the interval of interest is
empirical and consists of: determining a normalized Taylor signal
s(t) associated with the experimental Taylor signal S(t) by the
relation: s(t)=S(t)/S(t.sub.0), calculating the logarithm ln[s(t)],
determining the derivative .differential. ln s .differential. x
##EQU00050## relative to the variable x=(t-t.sub.0).sup.2, and
determining the values of the parameters .tau..sub.min and
.tau..sub.min related to the extrema of the derivative according to
the relations: .tau. min = a min ( .differential. ln s
.differential. x max ) - 1 / 2 ##EQU00051## .tau. max = a max (
.differential. ln s .differential. x min ) - 1 / 2 ##EQU00051.2##
with .alpha..sub.min=0.1; .alpha..sub.max=3, determining the
minimum G.sub.min and maximum G.sub.max using the equations: for
c=1: G.sub.min =.tau..sub.max.sup.-2,
G.sub.max=.tau..sub.min.sup.-2, for c=1: G.sub.min
=.tau..sub.min.sup.-2, G.sub.max=.tau..sub.max.sup.-2, for
c=-1/d.sub.f=-(1+a/3: G min = .tau. min ( 6 1 + a ) , G max = .tau.
max ( 6 1 + a ) . ##EQU00052##
10. Method according to claim 1, comprising a step of measuring an
average in T, G.sub.T, of the parameter G.sub.(1) based on the
experimental Taylor signal and/or an average in .GAMMA.,
G.sub..GAMMA., of the parameter G.sub.(1) based on the breakdown,
whereby each average may be used in a constraint that must be
observed by the amplitude distribution P(G.sub.(1)) that is the
solution of the foregoing equation.
11. Method according to claim 7, wherein the step of determining
the values of the minimum G.sub.min and maximum G.sub.max uses the
following equations: .beta. = 1 3 ln G .GAMMA. + 2 3 ln G T
##EQU00053## .gamma. = 2 3 ln G .GAMMA. G T ##EQU00053.2## then
G.sub.min=exp(.beta.-k {square root over (2)}.gamma.) and
G.sub.max=exp(.beta.+k {square root over (2)}.gamma.).
12. Method according to claim 1, comprising a step of determining a
peak time of the experimental Taylor signal S(t), comprising the
following sub-steps: obtaining a first estimate t.sub.0,guess of
the peak time, carrying out, for several different peak times to be
tested t.sub.0,i selected around the estimated peak time
t.sub.0,guess, a series of cumulant analyses considering several
ranges of time of different lengths specified based on a cutoff
level; selecting an optimal peak time to for which the first
cumulant .GAMMA..sub.1, the second cumulant .GAMMA..sub.2, and/or
the square of the ratio of the second cumulant to the first
cumulant diverges towards positive values when the cutoff level
increases, for which, respectively, the first cumulant
.GAMMA..sub.1, the second cumulant, .GAMMA..sub.2 and/or the square
of the ratio of the second cumulant to the first cumulant diverges
towards negative values when the cutoff level increases.
13. Data storage medium comprising instructions for the execution
of a method for determining the hydrodynamic ray, diffusion
coefficient, or molar mass distribution of a mixture of molecule or
particle species according to claim 1, when the instructions are
executed by a computer.
14. System configured to determine the hydrodynamic ray, diffusion
coefficient, or molar mass distribution of a mixture of molecule or
particle species including a computer, whereby the computer is
programmed to execute a method for determining the hydrodynamic
ray, diffusion coefficient, or molar mass distribution of a mixture
of molecule or particle species according to claim 1.
15. Method according to claim 10, wherein the step of determining
the values of the minimum G.sub.min and maximum G.sub.max uses the
following equations: .beta. = 1 3 ln G .GAMMA. + 2 3 ln G T
##EQU00054## .gamma. = 2 3 ln G .GAMMA. G T ##EQU00054.2## then
G.sub.min=exp(.beta.-k {square root over (2)}.gamma.) and
G.sub.max=exp(.beta.+k {square root over (2)}.gamma.).
Description
FIELD OF THE INVENTION
[0001] This invention concerns methods for determining the size
distribution of a mixture of particles by implementing the Taylor
dispersion and associated system, including the following steps:
[0002] injecting a sample of the mixture to be analyzed inside a
capillary in which an eluent is flowing; [0003] transporting the
sample injected along the capillary from an injection section to a
detection section thereof, in experimental conditions suitable to
generate a Taylor dispersion phenomenon that is measurable at the
level of the detection section; [0004] generating, by means of a
suitable sensor included in the detection section, a signal
characteristic of the Taylor dispersion of the transported sample;
[0005] acquiring the detection signal in order to obtain an
experimental Taylor signal; and [0006] analysing the experimental
Taylor signal.
[0007] Below, `particle` refers to any molecule in solution and/or
particles in suspension in the mixture.
[0008] In this document, a species includes all particles
characterized by the same size, e.g., the same hydrodynamic ray. A
species is thus associated with a `particle size` value.
BACKGROUND OF THE INVENTION
[0009] In this field, the `deconvolution` of a Taylor signal refers
to the processing of the experimental Taylor signal leading to the
determination of the hydrodynamic ray of each of the species
forming the mixture and the determination of the concentration of
each of these species.
[0010] The international application published under no. WO 2010
009907 A1 discloses a method of the aforementioned type, the
analysis step of which implements various deconvolution algorithms
for an experimental Taylor signal. However, these algorithms may
only be used in the specific case of a binary mixture, i.e., a
mixture of two species. Accordingly, these known algorithms do not
allow for the analysis of any desired sample, but only of samples
of which it is known in advance that they result from the mixture
of two species.
SUMMARY OF THE INVENTION
[0011] In practice, it is currently considered impossible to solve
the general problem of deconvoluting the experimental Taylor signal
of a sample of any given mixture of species.
[0012] The invention thus seeks to alleviate this problem by
proposing, in particular, a method for real-time analysis of an
experimental Taylor signal of a sample of any given mixture.
[0013] To this end, the invention concerns a method for determining
the size distribution of a mixture of molecule or particle species
including the following steps: [0014] injecting a sample of the
mixture to be analyzed inside a capillary in which an eluent is
flowing; [0015] transporting the sample injected along the
capillary from an injection section to a detection section thereof,
in experimental conditions suitable to generate a Taylor dispersion
phenomenon that is measurable at the level of the detection
section; [0016] generating, by means of a suitable sensor included
in the detection section, a signal characteristic of the Taylor
dispersion of the transported sample; [0017] processing the
detection signal in order to obtain an experimental Taylor signal
S(t); and [0018] analysing (200) the experimental Taylor signal
S(t),
[0019] characterized in that the step of analysing an experimental
Taylor signal S(t) of a sample of the mixture consists of searching
an amplitude distribution P(G.sub.(c)) that allows the experimental
Taylor signal S(t) to be broken down into a sum of Gaussian
functions by means of the equation:
{circumflex over
(S)}(t).ident..intg..sub.0.sup..infin.P(G.sub.(c))G.sub.(c).sup.c/2exp[-(-
t-t.sub.0).sup.2G.sub.(c).sup.c]dG.sub.(c)
where
[0020] t is a variable upon which the experimental Taylor signal
depends and t.sub.0 is a value of the variable t common to the
various Gaussian functions and corresponding to the peak of the
experimental Taylor signal S(t);
[0021] G.sub.(c) is a characteristic parameter of a Gaussian
amplitude function P(G.sub.(c)) and is associated:
[0022] where c=1, with the diffusion coefficient D of a species
according to the relation G.sub.(1)=12D/(R.sub.c.sup.2t.sub.0)
[0023] for c=-1, to the hydrodynamic ray R.sub.h of a species
according to the relation
G ( - 1 ) = 2 k B T .pi..eta. R c 2 t 0 R h - 1 ; ##EQU00001##
[0024] and
[0025] for c=-1/d.sub.f=-(1-a)/3, to the molar mass M of a species
according to the relation
G = 2 k B T .pi..eta. R c 2 t 0 ( 10 .pi. N a 3 K ) 1 / 3 M - ( 1 +
a 3 ) , ##EQU00002##
[0026] where k.sub.B is the Boltzmann constant, T is the absolute
temperature expressed in Kelvins at which the experiment is
conducted, .eta. is the viscosity of the eluent used, R.sub.c is
the internal ray of the capillary used, N.sub.a is Avogadro's
number, and K and a are Mark Houwink coefficients,
[0027] by implementing a constrained regularization algorithm
consisting of minimising a cost function H.sub..alpha. including at
least one constraint term associated with a constraint that must
observe the amplitude distribution P(G.sub.(c)) that is the
solution of the foregoing equation, whereby the minimization is
carried out on an interval of interest of the values of the
parameter G.sub.(c).
[0028] According to specific embodiments, the method includes one
or more of the following characteristics, taken alone or in all
combinations technically possible: [0029] the foregoing equation is
discretized by subdividing the interval of values of the parameter
G.sub.(c), whereby each discretization point G.sub.m is indexed by
an integer m that varies between the unit value and the value N,
whereby the point G.sub.m is at a distance from the point G.sub.m-1
corresponding to a sub-interval of the length c.sub.m; [0030] the
cost function takes the form:
[0030] H.sub..alpha.=.chi..sup.2+.alpha..sup.2.DELTA..sup.2
Where:
[0031] the first term .chi..sup.2 is a distance term between the
experimental Taylor signal S(t) and a reconstructed Taylor signal
defined by:
[0031] S(t)=.SIGMA..sub.m=1.sup.Nc.sub.mP(G.sub.m) {square root
over (G.sub.m)}exp[-(t-t.sub.0).sup.2G.sub.m], and [0032] the
second term .DELTA..sup.2 is a constraint term associated with the
at least one constraint that must observe the amplitude
distribution P(G) that is the solution of the foregoing equation,
whereby the second term is introduced by a Lagrange coefficient
.alpha., allowing the contribution of the second term of the cost
function H.sub.a to be adapted to the first term; [0033] the first
term .chi..sup.2 is a distance of the type `least squares` taking
the form:
[0033] .chi..sup.2=.SIGMA..sub.k=1.sup.L(S'(t.sub.k)-{circumflex
over (S)}(t.sub.k)).sup.2
[0034] where the experimental Taylor signal S(t)and the
reconstructed function S'(t)are sampled over time, whereby each
sample is indexed by an integer k varying between the unit value
and the value L; [0035] the at least one constraint that must be
observed by the amplitude distribution P(G) that is the solution of
the foregoing equation is a regularity constraint associated with a
constraint term .DELTA..sup.2 preferably taking the form:
[0035]
.DELTA..sup.2=.SIGMA..sub.m=2.sup.N-1[P(G.sub.m-1)-2P(G.sub.m)+P(-
G.sub.m+1)].sup.2; [0036] the analysis step includes a step of
determining the optimal value .alpha..sub.0 of the Lagrange
coefficient .alpha. such that the value of the distance term
.chi..sup.2 corresponding to the minimum of the cost function
H.sub..alpha.=.alpha.0 is close by values lower than a statistical
error .nu., preferably of the form .nu.=L-N; [0037] because the
interval of interest of the values of the parameter G.sub.(c) are
delimited by a minimum G.sub.m in and a maximum G.sub.max, the
method includes a step of determining the values of the minimum and
maximum; [0038] it includes a step of breaking down a normalized
Taylor signal s(t) associated with the experimental Taylor signal
S(t) into components by the relation: s(t)=S(t)/S(t.sub.0),
consisting of adjusting to the curve ln[s(t)] a second-order
polynomial of the variable (t-t.sub.0).sup.2 so as to determine the
first-order components .GAMMA..sub.1 and second-order components
.GAMMA..sub.2, and in that the step of determining the values of
the minimum G.sub.min and maximum G.sub.max uses the equations:
[0038] .beta. = ln .GAMMA. 1 - ln ( 1 + .GAMMA. 2 .GAMMA. 1 2 ) and
.gamma. = ln ( 1 + .GAMMA. 2 .GAMMA. 1 2 ) ##EQU00003##
[0039] where .beta. and .gamma. are respectively the average and
the standard deviation of the logarithm of the parameter G of a
log-normal distribution, followed by,
G.sub.min=exp(.beta.-k {square root over (2)}.gamma.) and
G.sub.max=exp(.beta.+k {square root over (2)}.gamma.); [0040] the
step of determining the values of the minimum and maximum of the
interval of interest is empirical and consists of: [0041]
determining a normalized Taylor signal s(t) associated with the
experimental Taylor signal S(t) by the relation:
s(t)=S(t)/S(t.sub.0), [0042] calculating the logarithm ln[s(t)],
[0043] determining the derivative
[0043] .differential. ln s .differential. x ##EQU00004##
relative to the variable x=(t-t.sub.0).sup.2, and [0044]
determining the values of the parameters .tau..sub.min and
.tau..sub.min related to the extrema of the derivative according to
the relations:
[0044] .tau. min = a min ( | .differential. ln s .differential. x |
max ) - 1 / 2 ##EQU00005## .tau. max = a max ( | .differential. ln
s .differential. x | min ) - 1 / 2 ##EQU00005.2##
[0045] with .alpha..sub.min=0.1; .alpha..sub.max=3, [0046]
determining the minimum G.sub.min and maximum G.sub.max using the
equations:
[0046] for c=1: G.sub.min=.tau..sub.max.sup.-2,
G.sub.max=.tau..sub.min.sup.-2,
for c=-1: G.sub.min=.tau..sub.min.sup.-2,
G.sub.max=.tau..sub.max.sup.-2,
for c=-1/d.sub.f=-(1+a)/3:
G min = .tau. min ( 6 1 + a ) , G max = .tau. max ( 6 1 + a ) ;
##EQU00006## [0047] it includes a step of measuring an average in
T, G.sub.T, of the parameter G.sub.(1) based on the experimental
Taylor signal and/or an average in .GAMMA., G.sub..GAMMA., of the
parameter G.sub.(1) based on the breakdown, whereby each average
may be used in a constraint that must be observed by the amplitude
distribution P(G.sub.(1)) that is the solution of the foregoing
equation; [0048] the step of determining the values of the minimum
G.sub.min and maximum G.sub.max uses the equations:
[0048] .beta. = 1 3 ln G .GAMMA. + 2 3 ln G T ##EQU00007## .gamma.
= 2 3 ln G .GAMMA. G T ##EQU00007.2##
[0049] then
G.sub.min=exp(.beta.-k {square root over (2)}.gamma.) and
G.sub.max=exp(.beta.+k {square root over (2)}.gamma.).
[0050] The invention also concerns a data storage medium including
instructions for the execution of a method for determining the
hydrodynamic ray, diffusion coefficient, or molar mass distribution
of a mixture of molecule or particle species as defined above when
the instructions are executed by a computer.
[0051] The invention lastly concerns a system for determining the
hydrodynamic ray, diffusion coefficient, or molar mass distribution
of a mixture of molecule or particle species including a computer,
whereby the computer is programmed to execute a method for
determining the hydrodynamic ray, diffusion coefficient, or molar
mass distribution of a mixture of molecule or particle species as
defined above.
BRIEF DESCRIPTION OF THE DRAWINGS
[0052] Other characteristics and advantages of the invention will
become apparent from the following detailed description, provided
by way of example and by reference to the attached drawings, in
which:
[0053] FIG. 1 is a schematic representation of a system for
determining the size distirbution of a particle mixture;
[0054] FIG. 2 is a schematic block representation of the method for
determining the size distirbution of a particle mixture implemented
by the system of FIG. 1; and
[0055] FIG. 3-5 (A and B) are graphs showing the results of the
implementation of the method of FIG. 2 in the case of an equimassic
mixture of two samples of synthetic polymers: FIG. 3 shows the
accumulation of three repetitions of the experimental Taylor
signal; FIG. 4 shows the hydrodynamic ray distribution obtained by
implementing the method of FIG. 2 compared with that obtained by
steric exclusion chromatography and provided by the supplier of the
synthetic polymer samples; and FIG. 5A shows the adjustment of the
Taylor signal by the method of FIG. 2, and FIG. 5B shows the
breakdown of the experimental Taylor signal of FIG. 3 into its
components.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Experimental Device
[0056] By reference to FIG. 1, the system for determining the size
of a mixture of particles 2 includes an experimental device 3
suited to generate a Taylor dispersion phenomenon and to generate
an experimental Taylor signal, and an analysis device 5 suited to
analyse the experimental Taylor signal output by the experimental
device 3 in order to determine in real time the size distribution
of the particle mixture, a sample of which was injected into the
experimental device.
[0057] The experimental device 3 includes, as is known, a capillary
6.
[0058] The experimental device 3 includes, in the vicinity of one
end of the capillary 6, an injection section 7, and, in the
vicinity of the other end of the capillary 6, a detection section
9.
[0059] The injection section 7 includes means 11 for injection into
the capillary 6 of a sample of the mixture to be analyzed. The
injection section 7 also includes means to allow an eluent to flow
inside the capillary 6 from the injection section 7 to the
detection section 9. These flow means are shown schematically in
FIG. 1 by block 13.
[0060] The detection section 9 is optical. It is equipped with an
optical cell including a light source S and an optical system 15
suited to cause the rays of light emitted by the source S to
converge on a narrow portion of the capillary 6. Along the optical
axis of the optical system 15, but opposite the lighted side of the
capillary 6, the cell includes a CCD, diode array, or
photomultiplier sensor 17 suited to collect the light that has
passed through the capillary 6 and to generate a detection signal
corresponding to the light collected. The sensor 17 is electrically
connected to an electronic card 19 for pre-processing and
digitising the detection signal generated by the sensor 17. The
card 19 outputs a digital measurement signal that is
time-dependent. This measurement signal is referred to as the
`experimental Taylor signal` or Taylorgram. It is indicated by the
notation S(t) in the following. It depends on the time t.
[0061] The experimental Taylor signal S(t) is sampled at a
predetermined temporal frequency, although the sampling points
t.sub.k(k=1, . . . , L) are at regular intervals. The experimental
Taylor signal S(t) thus consists of a group of L pairs of data
(t.sub.k,S(t.sub.k)).
[0062] In one variant, the detection section may be of another
type, e.g., a conductivity detector, using mass spectrometry,
fluorescence (laser-induced, if applicable), electrochemical, light
diffusion, or more generally, any type of detector used in
capillary electrophoresis. In particular, the Taylor signal may not
be a time signal (scrolling the Taylor peak in front of a narrow
sensor), but rather a spatial signal (instantaneous capture of the
Taylor peak in front of an extended sensor). In this case, the
variable t does not represent time, but the position along the
capillary.
[0063] The analysis device 5 consists of a computer including an
input/output interface 21 to which the electronic card 19 of the
experimental device 3 is connected.
[0064] The computer further includes a memory 23, such as a RAM
and/or ROM, as well as a processing unit 25, such as a
microprocessor. The computer also includes human-machine interface
means, indicated by the number 27 in FIG. 1. They include, e.g., a
touch screen allowing a user to interact with the computer 5. All
of the components of the computer are interconnected in a known
fashion, e.g., by a data exchange bus.
[0065] The experimental Taylor signal S(t) is processed by a
software application, the instructions of which are stored in the
memory 23 and executed by the processing unit 25. This software is
shown schematically in FIG. 1 by block 31.
Modelling of the Taylor Signal
[0066] In the known manner, assuming that i) the contribution of
the diffusion along the axis of the capillary 6 to the dispersion
at the level of the peak of the signal is negligible, ii) the
injection time of the sample into the capillary 6 is sufficiently
short (typically, the injected volume is less than 1% of the volume
of the capillary), and iii) the detection device is sensitive to
the mass of the molecules, the real Taylor signal S(t) of a
monodispersed
[0067] sample, i.e., a sample having only one species, and,
accordingly, characterized by a hydrodynamic ray value R.sub.h or
diffusion coefficient D, is modelled by a Gaussian function:
S(t)=CM.rho. {square root over
(D)}exp[-(t-t.sub.0).sup.212D/(R.sub.c.sup.2t.sub.0)]+B. (1.)
Where:
[0068] C is an instrumental constant,
[0069] M is the molar mass of the species,
[0070] .rho. is the molar concentration of the species,
[0071] R.sub.c is the internal ray of the capillary,
[0072] t.sub.0 is the moment corresponding to the peak of the
Taylor signal, and
[0073] B is an offset constituting a measurement artifact (this
term will be omitted in the following for clarity and because it is
taken into account in the constrained regularization method).
[0074] It should be emphasized that assumption iii) depends on the
nature of the sensor used in the detection section, and that the
use of another type of sensor results in modifications to the
equations shown herein in a manner known to persons skilled in the
art.
[0075] The real Taylor signal S(t) of a polydispersed sample, i.e.,
a sample including
[0076] several species, is modelled for the sum of the
contributions of each of the species. Thus, assuming a mixture
including a continuum of species, equation (1) is generalized by a
continuous sum of Gaussian functions according to:
S(t)=.intg..sub.0.sup..infin.CM(D).rho.(D) {square root over
(D)}exp[-(t-t.sub.0).sup.212D/(R.sub.c.sup.2t.sub.0)]dD, (2.)
[0077] In equation (2), the Gaussian functions are all centred on
the same reference time t.sub.0.
[0078] The parameter G=12D/(R.sub.C.sup.2T.sub.0) is introduced,
causing the equation (2) to become:
S(t)=.intg..sub.0.sup..infin.P(G) {square root over
(G)}exp[-(t-t.sub.0).sup.2G]dG (3.)
where P(G) is referred to in the following as the `amplitude
distribution` of the Guassian functions of the parameter G.
[0079] A value of the parameter G is associated, via the diffusion
coefficient D, with a species. For example, the
Stokes-Einstein-Sutherland formula allows for the association of a
value of the parameter G with the species characterized by the
hydrodynamic ray R.sub.h, according to the relation:
G = 2 k B T .pi..eta. R c 2 t 0 R h ( 4. ) ##EQU00008##
Where:
[0080] k.sub.B is the Boltzmann constant,
[0081] T is the absolute temperature expressed in Kelvins at which
the experiment is carried out, and
[0082] .eta. is the viscosity of the eluent used.
[0083] Thus, in equation (3), each Gaussian function P(G) {square
root over (G)}exp[-(t-t.sub.0).sup.2G] represents the contribution
of one species to the total amplitude of the real Taylor signal
S(t). The amplitude P(G) of each Gaussian function depends directly
on the concentration of the corresponding species in the
mixture.
[0084] The function of the software 31 is to determine the
distribution P(G) that is the solution of the following equation
corresponding to equation 83) when the real Taylor signal S(t) is
replaced with the experimental Taylor signal S(t):
{circumflex over (S)}(t).ident..intg..sub.0.sup..infin.P(G) {square
root over (G)}exp[-(t-t.sub.0).sup.2G]dG (5.)
Measurement Error Introduced by the Experimental Device
[0085] Like any measurement device, the detection section 9
introduces a systematic measurement error such that the
experimental Taylor signal S(t) is not exactly equal to the real
Taylor signal S(t).
[0086] The solution of the equation (5) then results in the
determination of several distributions that are each solution to
the measurement error. In other words, the solution of the equation
(5) results in the identification of several families of Gaussian
functions that each result in sum in a reconstructed Taylor signal
S(t)=.intg..sub.0.sup..infin.P(G) {square root over
(G)}exp[-(t-t.sub.0).sup.2G]dG that is adjusted to the experimental
Taylor signal S(t), with the adjustment criterion taking into
account the measurement error introduced by the detection
section.
[0087] However, amongst the various distributions P(G) that are
solutions to the equation (5), only some have physical
significance. It is such a `physical` solution that the software 31
is suited to determine.
Method for Determining the Physically Significant Distribution
[0088] To solve this problem, the software 31 uses an algorithm
that implements a constrained regularization method.
[0089] This algorithm is based on the following equation, which
results from a discretization relative to the parameter G of the
equation (5):
{circumflex over
(S)}(t).ident..SIGMA..sub.m=1.sup.N-1c.sub.mP(G.sub.m) {square root
over (G.sub.m)}exp[-(t-t.sub.0).sup.2G.sub.m] (6.)
where the interval of the values of interest of the parameter G,
between the predetermined minimum G.sub.min=G.sub.0 and maximum
G.sub.max=G.sub.N, is subdivided into N sub-intervals identified by
the integer m of the length c.sub.m. Preferably, the various
sub-intervals have the same length:
c m = G N - G O N ##EQU00009##
[0090] The limits are such that the interval on which equation (6)
is discretized exceeds the interval for which the distribution P(G)
is not nil. Accordingly, P(G.sub.1)=P(G.sub.N)=0.
[0091] The unknowns of equation (6) consist of all amplitudes
P(G.sub.m), m=1, . . . N.
[0092] In principle, the solution of equation (6) is obtained by a
process of adjusting the experimental Taylor signal S(t) by the
reconstructed Taylor signal
S(t)=.SIGMA..sub.m=1.sup.Nc.sub.mP(G.sub.m) {square root over
(G.sub.m)}exp[-(t-t.sub.0).sup.2G.sub.m]. That is, for any time
t.sub.k, S'(t.sub.k) must be as near as possible to S(t.sub.k).
[0093] In order to obtain robust results that are physically
significant, however, it is necessary to take into account all
information available on the amplitudes P(G.sub.m) during
[0094] the adjustment process so as to reject all solutions that
are not physically acceptable.
[0095] To this end, the constrained regularization algorithm solves
equation (6) by minimising a cost function dependent on the N
unknown P(G.sub.m) and translating the
[0096] information available on the amplitudes P(G.sub.m) into
constraints.
[0097] In the current embodiment, the cost function H.sub..alpha.
takes the following form:
H.sub..alpha.=.chi..sup.2+.alpha..DELTA..sup.2 (7.)
[0098] It includes a first term .chi..sup.2 corresponding to a
`distance` between the experimental Taylor signal S(t.sub.k) and a
reconstructed Taylor signal S'(t.sub.k).
[0099] For example, this first term is a distance of the type
`least squares`:
.chi..sup.2=.SIGMA..sub.k=1.sup.L(S'(t.sub.k)-{circumflex over
(S)}(t.sub.k)).sup.2 (8.)
[0100] Another distance measure may be used, in particular, one
that, in the foregoing sum, weights each term by a coefficient wk
inversely proportional to the noise affecting the measurement taken
at the instant t.sub.k.
[0101] The cost function H.sub..alpha. includes a second term
.DELTA..sup.2, referred to as the constraint term, expressing a
constraint that penalises the amplitudes P(G.sub.m) that have no
physical significance.
[0102] For example:
.DELTA..sup.2=.SIGMA..sub.m=2.sup.N-1[P(G.sub.m-1)-2P(G.sub.m)+P(G.sub.m-
+1)].sup.2 (9.)
[0103] In this example, the constraint term corresponds to the sum
of the terms elevated to the square of the second derivative of the
distribution P(G). The constraint term translates a regularity
constraint. The amplitudes P(G.sub.m) that vary too rapidly
relative to their neighbours, P(G.sub.m-1) or P(G.sub.m+1), are
thus penalised.
[0104] Another example of a regularity constraint is the
following:
.DELTA..sup.2=.SIGMA..sub.m=3.sup.N-1[P(G.sub.m-2)-3P(G.sub.m-1)+3P(G.su-
b.m)-P(G.sub.m+1)].sup.2 (9.)
[0105] This constraint term corresponds to the sum of the terms
elevated to the square of the third derivative of the distribution
P(G). The generalization to a regularity constraint based on the
n-th derivative (n.gtoreq.1) is immediate and can be carried out by
a person skilled in the art.
[0106] In the following, it is assumed that the regularity
constraint used is that of the second derivative of the
distribution P(G), eq. (9).
[0107] The first and second terms of the cost function
H.sub..alpha. have relative contributions that may be adapted by
selecting the value of a coefficient .alpha., a Lagrange
coefficient. This coefficient verifies the size of the constraint
term relative to the distance term. If .alpha. is very small, the
constraint term is negligible. In this case, the minimization of
the cost function yields the same result as a simple adjustment to
the experimental data. For values of .alpha. that are too large, on
the other hand, a significant cost is assigned to the constraint on
the P(G), and the algorithm will reject the solutions that do not
observe the constraint at the risk of preserving a solution that
does not properly fit the experimental data.
[0108] Supplemental constraints that may be expressed in the form
of supplemental equalities or inequalities, or P(G.sub.m) linear
inequalities are directly imposed during the search for the minimum
of the cost function by limiting the search to the subregions
specific to the space of the P(G.sub.m).
[0109] For example, the constraint that the amplitudes are
positive, P(G.sub.m).gtoreq.0 .A-inverted.m , is imposed by
minimising the cost function only on the half space of the positive
amplitudes.
[0110] For example, if the value of an average G of the parameter G
is determined, a supplemental constraint on the amplitudes
P(G.sub.m) that are solutions to the equation (6) is that the
amplitudes allow the previously determined average G to be
determined at a deviation .epsilon.. This constraint is also
expressed in the form of P(G.sub.m) linear inequalities.
G-.epsilon..ltoreq..SIGMA..sub.m=1.sup.NP(G.sub.m)G.sub.m.ltoreq.G+.epsi-
lon. (11)
[0111] With, for example: .epsilon./G=5%.
[0112] Equation 12 may easily be generalized to other types of
averages than the arithmetical average:
G=.SIGMA..sub.m=1.sup.NP(G.sub.m)G.sub.m (12)
[0113] for example the averages G.sub.T and G.sub..GAMMA. which
will be introduced below.
[0114] A crucial point is the choice of the coefficient .alpha. of
the cost function H.sub..alpha. in equation 7. Two strategies may
be used to select the coefficient .alpha.: [0115] A first strategy
consists of choosing the greater value .alpha. such that the value
of the distance term .chi..sup.2 does not exceed the statistically
expected value that depends on the measurement error and the number
of degrees of freedom in the constrained regularization process.
Thus, if, for each experimental point, the standard deviation of
the measurement error .sigma..sub.k, is known, .alpha. is selected
such that the normalized value .chi..sub.norm.sup.2 of the distance
term .chi..sup.2 does not exceed the number of degrees of freedom:
.nu.=L-N, where .chi..sub.norm.sup.2 is given Dv tne equation:
[0115] .chi. norm 2 = .SIGMA. k = 1 N ( S ' ( t k ) - S ^ ( t k ) )
2 .sigma. k 2 ( 10. ) ##EQU00010##
[0116] If the standard deviation of the noise .sigma..sub.k is not
known, an estimated value .sigma..sub.est thereof may be determined
based on the mean deviation between the experimental data and the
best possible adjustment without taking into account the
constraint, i.e., that obtained for .alpha.=0,:
.sigma. est 2 = 1 N .SIGMA. k = 1 N ( S ' ( t k ) ( .alpha. = 0 ) -
S ^ ( t k ) ) 2 , ( 11. ) ##EQU00011##
where S'(t.sub.k)(.alpha.=0) is the value of the k-th point of the
Taylor signal reconstructed based on the amplitudes P(G.sub.m)
obtained by minimising only the first term of the cost function
H.sub..alpha.=0.
[0117] Once the noise has been estimated, .alpha. is selected such
that the normalized value .chi..sub.norm.sup.2 of tne distance term
.chi..sup.2, calculated by replacing .sigma..sub.k with
.sigma..sub.est in equation 12 does not exceed the number of
degrees of freedom: .nu.=L-N
[0118] A second strategy consists of selecting .alpha. value for a
that gives equal weight to the distance term and the constraint
term.
[0119] In this case, the parameter .alpha. that is retained is the
one for which, once the constraint function H.sub..alpha. has been
minimized, the result is .chi..sup.2=.alpha..DELTA..sup.2.
[0120] In practice, the selection of .alpha. is made then by
scanning a large range of values of the coefficient .alpha.. For
each value of .alpha., all amplitudes P(G.sub.m) that minimise the
cost function H.sub..alpha. are determined. The corresponding
values of .chi..sub.norm.sup.2 , H.sub..alpha. and P(G.sub.m) are
recorded. Amongst all of the trials, the Lagrange coefficient
.alpha..sub.0 having the greatest value such that
.chi..sub.norm.sup.2.ltoreq..nu. is finally retained.
[0121] In order to improve the efficiency of the digital search,
the values of .alpha. are first scanned on a trial grid with large
steps, and then the value of .alpha. is refined using a finer
grid.
Determining G.sub.T and G.sub..GAMMA. from an Experimental Taylor
Signal
[0122] Below, two averages of G are presented that may be obtained
directly from the experimental Taylor signal.
[0123] The T average of the parameter G is defined as follows:
G T = .SIGMA. m = 1 N c m P ( G m ) .SIGMA. m = 1 N c m P ( G m ) G
m = 1 [ G - 1 _ ] ( 12. ) ##EQU00012##
and the .GAMMA. average of the parameter G is defined as
follows:
G .GAMMA. = .SIGMA. m = 1 N c m P ( G m ) G m 3 / 2 .SIGMA. m = 1 N
c m P ( G m ) G m 1 / 2 , ( 13. ) ##EQU00013##
where c.sub.m are those defined in relation to equation (6).
[0124] The purpose of determining these averages is twofold: [0125]
it allows constraints on the amplitudes P(G.sub.m) to be added
during the solution of equation (6); [0126] it allows for an
estimation of the average and breadth of the distribution P(G).
This information is not only interesting in itself, but also allows
for a determination of the interval of the values of the parameter
G on which a solution to equation (6) can be sought.
[0127] G.sub.T and G.sub..GAMMA. may be calculated respectively
based on the temporal variance of the experimental Taylor signal
and based on the cumulative approach described below.
Determining G.sub.T Based on the Temporal Variance of the
Experimental Taylor Signal
[0128] It is shown that:
.intg. 0 + .infin. S ( t ) t .intg. 0 + .infin. S ( t ) ( t - t 0 )
2 t .apprxeq. .intg. - .infin. + .infin. S ( t ) t .intg. - .infin.
+ .infin. S ( t ) ( t - t 0 ) 2 t = .pi. .intg. 0 + .infin. P ( G )
G .pi. 2 .intg. 0 + .infin. G - 1 P ( G ) G = 2 [ G - 1 _ ] - 1 = 2
G T ( 14. ) ##EQU00014##
[0129] Thus, the T average of the parameter G is accessible by
integrating the experimental Taylor signal.
[0130] With G=12D/(R.sub.c.sup.2t.sub.0), the T average of the
parameter D is given by:
D T = R c 2 t 0 24 .intg. 0 + .infin. S ( t ) t .intg. 0 + .infin.
S ( t ) ( t - t 0 ) 2 t ( 15. ) ##EQU00015##
Determining G.sub..GAMMA. Based on a Breakdown of the Experimental
Taylor Signal
[0131] In this part, the breakdown of the experimental Taylor
signal is described in the case of a sample that is moderately
polydispersed.
[0132] It is assumed that the size distribution is discrete.
Equation (2) then becomes:
S(t)=C'.SIGMA..sub.i=1.sup.N.rho..sub.iM.sub.i {square root over
(D)}.sub.iexp[-(t-t.sub.0).sup.212D.sub.i/(R.sub.c.sup.2t.sub.0)]
(16.)
where .eta..sub.i is the molar concentration of the i-th species in
the mixture, and M.sub.i and D.sub.i are respectively the molar
mass and diffusion coefficient thereof.
[0133] It is useful to `normalise`the Taylor signal relative to the
height of its peak by introducing:
s(t)=S(t)/S(t.sub.0=.SIGMA..sub.i=1.sup.Nf.sub.iexp[-(t-t.sub.0).sup.2G.-
sub.i] (17.)
Where:
[0134] -G.sub.i=12D.sub.i/(R.sub.c.sup.2i.sub.0)and
[0135] -f=.rho..sub.iM.sub.i {square root over
(D)}.sub.i/.SIGMA..sub.i=1.sup.N(.rho..sub.iM.sub.i {square root
over (D)}) is the relative contribution of the i-th species in
[0136] the Taylor signal. It should be noted that t, depends on the
diffusion coefficient of the i-th species.
[0137] The .GAMMA. average of the parameter G is then expressed as
follows:
G .GAMMA. .ident. i = 1 N .rho. i M i D i G i i = 1 N .rho. i M i D
i = .SIGMA. i = 1 N f i G i . ( 18. ) ##EQU00016##
[0138] By positing G.sub.i=G.sub..GAMMA.=.delta.G.sub.i, with
.delta.G.sub..GAMMA.=0, equation (19) becomes:
s(t)=exp[-(t-t.sub.0).sup.2G.sub..GAMMA.].SIGMA..sub.i=1.sup.Nf.sub.i
exp[-(t-t.sub.0).sup.2.delta.G.sub.i], (19.)
which is the product of a Gaussian function by correction
terms.
[0139] If (t-t.sub.0).sup.2.delta.G.sub.i<<1, i.e., near the
peak of the Taylor signal, the limited development of the second
term of equation (21) results in:
exp[-(t-t.sub.0).sup.2.delta.G.sub.i]=1-(t-t.sub.0).sup.2.delta.G.sub.i+-
1/2[(t-t.sub.0).sup.2.delta.G.sub.i].sup.2+ . . . (20.)
[0140] Which leads, in equation (21), using
.SIGMA..sub.i=1.sup.Nf.sub.i=1 and
.SIGMA..sub.i=1.sup.N.delta.G.sub.i=0 to:
s(t)=exp[-(t-t.sub.0).sup.2G.sub..GAMMA.](1+1/2(t-t.sub.0).sup.4.delta.G-
.sup.2.sub..GAMMA.+ . . . ) (21.)
[0141] Where the size distribution is not too broad (i.e., a
slightly polydispersed sample), it is possible to express the
normalized Taylor signal s(t) as the sum of a Gaussian function (as
in the case of a monodispersed sample) and correction terms (to
take into account a deviation in the case of a monodispersed
sample).
[0142] Taking the logarithm of the expression (23) and carrying out
a new limited development, the following is obtained:
ln[s(t)]=-(t-t.sub.0).sup.2G.sub..GAMMA.+1/2(t-t.sub.0).sup.4.delta.G.su-
p.2.GAMMA.+ . . . (22.)
[0143] The equation (24) is the desired cumulant development. The
coefficients .GAMMA..sub.1=G.sub..GAMMA. and
.GAMMA..sub.2.delta.G.sup.2.sub..GAMMA. are the first- and
second-order cumulants of this development.
[0144] They may be obtained by adjusting a second-order polynomial
of the variable (t-t.sub.0).sup.2 to the function ln[s(t)].
[0145] The first cumulant also provides access to the .GAMMA.
average of the diffusion coefficient:
D .GAMMA. = G .GAMMA. R c 2 t 0 12 = .GAMMA. 1 R c 2 t 0 12 . ( 23.
) ##EQU00017##
[0146] The .GAMMA. average is different to the aforementioned T
average because the diffusion coefficient D appears there at
different powers. These two averages contain different information
on the distribution P(G). They allow for the addition of two
constraints in the cost function to be minimized.
[0147] The second cumulant is linked to the .GAMMA. average of the
variance of the distribution of the diffusion coefficients,
providing an estimate of the polydispersity of the sample. More
specifically, the ratio of the second cumulant divided by the
square of the first cumulant gives:
.GAMMA. 2 .GAMMA. 1 2 = .delta. G 2 .GAMMA. G .GAMMA. 2 = D 2
.GAMMA. - D .GAMMA. 2 D .GAMMA.` 2 = D 5 / 2 _ D 1 / 2 _ - ( D 3 /
2 _ D 1 / 2 _ ) 2 ( D 3 / 2 _ D 1 / 2 _ ) 2 . ( 24. )
##EQU00018##
Selection of the Interval of the Values of the Parameter G on Which
to Seek a Solution
[0148] In the constrained regularization adjustment procedure, the
choice of the interval of the values G on which the distribution
P(G) is sought is an important factor.
[0149] In fact, the number N of points used in the discretization
of the equation (6) cannot be too large; otherwise, the calculation
time for the adjustment will be too long.
[0150] Furthermore, N must be significantly smaller than the number
L of digitization points of the experimental Taylor signal.
[0151] Typical values of N are in the range of 50-200.
[0152] These considerations show that the interval [G.sub.min,
G.sub.max] (G.sub.min=G.sub.0 and G.sub.max=G.sub.N) must be
carefully chosen.
[0153] However, the interval [G.sub.min, G.sub.max] must be greater
than the interval on which the distribution P(G) is not nil in
order to avoid artifacts due to the truncation of this
distribution.
[0154] Additionally, if the interval on which the distribution P(G)
is not nil is a sub-interval of the interval [G.sub.min, G.sub.max]
that is too narrow, the details of the distribution P(G) will be
weekly resolved during discretization.
[0155] It is also essential to define an automatic procedure
allowing for the determination of G.sub.min and G.sub.max, such
that the user does not waste time selecting the limits of the
interval and avoiding a series of trial/error.
[0156] We propose three possible approaches to determine G.sub.min
and G.sub.max. The first two approaches are based on the
calculation of the equivalent log-normal distribution, whilst the
third is empirical and based on the representation of ln[S(t)]
depending on (t-t.sub.0).sup.2 in the same system of axes that is
used for the breakdown into cumulants.
Determination of the Interval Based on a Log-Normal
Distribution
[0157] Equivalent Log-Normal Distribution
[0158] A log-normal distribution often allows for a highly accurate
description of the size distribution of a polymer or particle
sample:
PDF ( G ) = 1 G .gamma. 2 .pi. exp [ - ( ln G - .beta. ) 2 2
.gamma. 2 ] ( 25. ) ##EQU00019##
Where:
[0159] PDF(G) is the probability density that the particles of the
sample will have a G value between G and G+dG ; and,
[0160] .beta. and .gamma. are respectively the average and the
standard deviation of the logarithm of the parameter G, ln G.
[0161] Although the log-normal distribution may be a poor model for
more complex mixtures, the determination of the equivalent
log-normal distribution of any mixture is useful to estimate the
interval of the values of the parameter G on which the distribution
P(G) that is the solution of equation (6) is to be sought.
[0162] The probability density PDF (G) depends only on the
parameters .beta. and .gamma.. It is possible to determine these
two parameters from G.sub.T and G.sub..GAMMA..
[0163] The definition is:
G T = [ .intg. 0 .infin. G - 1 PDF ( G ) G ] - 1 ( 26. ) G .GAMMA.
= .intg. 0 .infin. G 3 / 2 PDF ( G ) G .intg. 0 .infin. G 1 / 2 PDF
( G ) G ( 27. ) ##EQU00020##
[0164] By replacing equation (27) in equations (28) and (29), the
following is obtained:
.beta. = 1 3 ln G .GAMMA. + 2 3 ln G T ( 28. ) .gamma. = 2 3 ln G
.GAMMA. G T ( 29. ) ##EQU00021##
[0165] It is also possible to obtain the equivalent log-normal
distribution from first- and second-order cumulants according to
the following relations:
.beta. = ln .GAMMA. 1 - ln ( 1 + .GAMMA. 2 .GAMMA. 1 2 ) ( 30. )
.gamma. = ln ( 1 + .GAMMA. 2 .GAMMA. 1 2 ) ( 31. ) ##EQU00022##
[0166] In conclusion, it is possible to determine the log-normal
distribution either from G and G.sub..GAMMA. according to equations
(30) and (31) or from .GAMMA..sub.1 and .GAMMA..sub.2 using
equations (32) and (33).
Calculation of the Limits of the Interval Based on the Equivalent
Log-Normal Distribution
[0167] The G.sub.min et G.sub.max may be estimated by replacing the
distribution P(G) with an equivalent log-normal distribution.
[0168] The objective is for the interval [G.sub.min, G.sub.max] to
cover a significant fraction of the log-normal distribution
equivalent to the experimental Taylor signal. This fraction of the
distribution is yielded by:
.DELTA.Q.sub.G=Q.sub.G(G.sub.max)-Q.sub.G(G.sub.min), (32.)
where Q.sub.G(G) is the cumulative probability defined by:
Q.sub.G(G)=.intg..sub.0.sup.GdG'PDF(G'). (33.)
[0169] Furthermore, it is preferable for the interval
[Q.sub.G(G.sub.min), Q.sub.G(G.sub.max)] to be distributed
symmetrically relative to the median value Q.sub.G=1/2. Thus::
Q.sub.G(G.sub.min)=1-Q.sub.G(G.sub.max) (34.)
[0170] In the context of this assumption, equation (36) yields:
erf ( ln G max - .beta. 2 .gamma. ) = 2 Q G ( G max ) - 1 = .DELTA.
Q G ( 35. ) ##EQU00023##
where erf is the error function known to persons skilled in the
art.
[0171] This results in:
G.sub.max=.beta.+k {square root over (2)}.gamma. i.e.,
G.sub.max=exp(.beta.+k {square root over (2)}.gamma.);
G.sub.max=exp(.beta.+k {square root over (2)}.gamma.), (36.)
and
G.sub.min=.beta.-k {square root over (2)}.gamma. i.e.,
G.sub.min=exp(.beta.-k {square root over (2)}.gamma.);
G.sub.min=exp(.beta.-k {square root over (2)}.gamma.), (37.)
with k=erf.sup.-1(.DELTA.Q.sub.G) where erf.sup.1 is the inverse
error function.
[0172] For example, if .DELTA.Q.sub.G=99.53%, then k=2, or if
.DELTA.Q.sub.G=99.998%, then k=3.
Empirical Determination of the Interval Based on the Breakdown into
Cumulants
[0173] For the sake of simplicity, the following notation will be
used: x=(t-t.sub.0).sup.2.
[0174] For a monodispersed sample, ln[s(t)] as a function of x is a
line, the gradient of which gives the diffusion coefficient of the
species of the sample. That is:
| .differential. ln s .differential. x | = G ( 38. )
##EQU00024##
[0175] Thus, .differential.lns/.differential.x does not depend on
x, i.e., it is not time-dependent.
[0176] On the other hand, for a polydispersed sample, the curve
ln[s(t)] depending on x has a curve that is calculated by
determining the derivative .differential.lns/.differential.x. This
is proportional to the parameter G.
[0177] Based on equation (40), it is assumed that G.sub.min is
linked to the minimum of the local gradient of ln s (in absolute
value), because the species with low diffusion coefficients
correspond to low G values, and thus to a slight decrease of the
Taylor signal over time. Accordingly, it is assumed that:
G min = b min | .differential. ln s .differential. x | min ( 39. )
##EQU00025##
where the minimum of |.differential.lns/.differential.x| sought on
an adapted interval of x, to be determined empirically, and where
b.sub.min is a numerical coefficient, also to be determined
empirically.
[0178] By studying a large number of Taylor signals of samples of
all kinds, we have found that the suitable interval of x is that
for which the signal s(t) decreases by two decades. This
corresponds to an interval in time between the time t.sub.0
corresponding to the peak of s(t) and the time t.sub.1 such that
s(t.sub.1)=S(t.sub.1)/S(t.sub.0)=0.01.
[0179] Likewise, it is considered that:
G max = b max | .differential. ln s .differential. x | max ( 40. )
##EQU00026##
[0180] where the maximum of |.differential.lns/.differential.x| is
sought on the same interval of x.
[0181] In analysing an experimental Taylor signal, it is simpler to
estimate the characteristic decrease time of s(t). Defining the
decrease time as .tau.=G.sup.-1/2, minimum and maximum decrease
times are defined based on the aforementioned relations, according
to:
.tau. min = a min ( | .differential. ln s .differential. x | max )
- 1 / 2 ( 41. ) .tau. max = a max ( | .differential. ln s
.differential. x | min ) - 1 / 2 ( 42. ) ##EQU00027##
[0182] By studying a large number of Taylor signals of samples of
all kinds, we have found that the following values of the
parameters .alpha..sub.min and .alpha..sub.max are able to frame
the desired min and max values.
.alpha..sub.min=0.1; .alpha..sub.max=3 i.e., b.sub.min=1/9;
b.sub.max=100 (43.)
[0183] Lastly, the values G.sub.min, G.sub.max to use in order to
adjust the data are calculated according to:
G.sub.min=.tau..sub.max.sup.-2 (44.)
G.sub.max=.tau..sub.min.sup.-2 (45.)
Calculation of the Distributions According to the Parameter D,
R.sub.h, or M Based on the Distribution According to the Parameter
G
[0184] Once the distribution of the amplitudes P(G) has been
obtained, the distribution of the amplitudes P.sub.D(D) according
to the diffusion coefficient D, can be easily calculated.
[0185] The following equation links the probability distribution
P.sub.y of the stochastic variable y to the probability
distribution P.sub.x of the stochastic variable x, where x is a
function of y:
P y ( y ) = P x ( x ) | x = x ( y ) | .differential. x
.differential. y | x = x ( y ) ( 46. ) ##EQU00028##
[0186] with G=12D/(R.sub.c.sup.2t.sub.0), the following is
obtained:
P D ( D ) = 12 R c 2 t 0 P ( G ) | G = 12 D R c 2 t 0 .intg. 0
.infin. P ( G ) G ( 47. ) ##EQU00029##
[0187] In this expression, an integral was introduced to the
denominator because the distribution P(G) is not necessarily
normalized.
[0188] It is often desirable to express the polydispersity of the
sample in terms of amplitude distribution according to the first
hydrodynamic ray R.sub.h, or the parameter of molar mass, M.
[0189] These two distributions may be calculated based on the
distribution P(G) using equation (46) and the following
transformation rules:
G = 2 k B T .pi..eta. R c 2 t 0 R h - 1 ( 48. ) G = 2 k B T
.pi..eta. R c 2 t 0 ( 10 .pi. N a 3 K ) 1 / 3 M - ( 1 + a 3 ) ( 49.
) ##EQU00030##
[0190] Equation (48) uses the Stokes-Einstein relation
D = k B T 6 .pi..eta. R h , ##EQU00031##
[0191] where k.sub.B is the Boltzmann constant, T the absolute
temperature, and .eta. the viscosity of the eluent.
[0192] Equation (49) uses the Einstein equation for the viscosity
of a diluted suspension and the Mark Houwink equation linking the
intrinsic viscosity [.eta.] to the molar mass according to the
relation:
[.eta.]=K M.sup.a (50.)
[0193] where K and a are the Mark Houwink coefficients.
[0194] The following relation, which gives the hydrodynamic ray as
a function of molar mass, may also be used:
R h = ( 3 K 10 .pi. N a ) 1 / 3 M ( 1 + a 3 ) ( 51. )
##EQU00032##
[0195] where N.sub.a is Avogadro's number and d.sub.f=3/(1+a) is
the fractal dimension of the object (e.g., d.sub.f=3 for an
ordinary compact object, 2 for a statistical polymer, and 5/3 for a
polymer in a good solvent).
[0196] Equations (48) and (49) show that G is non-linear in R.sub.h
and M, respectively, whilst G is simply proportional to D.
[0197] Due to this non-linearity, the transformation of the
distribution according to the parameter G identified as the
solution of equation (6) results in a distribution according to the
parameter R.sub.h, or the parameter M, which does not necessarily
observe the constraint of the cost function, in particular the
regularity constraint. In most cases, the transform results in fact
in the presence of non-physical peaks or oscillations in the
distribution according to the parameter R.sub.h, or the parameter
M.
[0198] This is why, in a variant of the method described in detail
above, the method consists of directly seeking the distribution
according to the parameter R.sub.h, or the parameter M, which
observes the constraint(s) and allows for the correct reproduction
of the experimental data, by constrained regularization.
[0199] To this end, the experimental Taylor signal is broken down
on a family of Gaussians of an adapted parameter: The equation (5)
is thus generalized in the form of:
{circumflex over
(S)}(t).ident..intg..sub.0.sup..infin.P(G.sub.(c))G.sub.(c).sup.c/2exp[-(-
t-t.sub.0).sup.2G.sub.(c).sup.c ]dG, (52.)
where the three following cases are considered:
[0200] 1) c=1: to be used when seeking the amplitude distribution
according to D;
[0201] 2) c=-1: to be used when seeking the amplitude distribution
according to R.sub.h;
[0202] 3) c=-1/d.sub.f=-(1+a)/3: to be used when seeking the
amplitude distribution according to M;
[0203] P.sub.norm(G.sub.(c)) is the distribution P(G.sub.(c)) ,
properly normalized:
P norm ( G ( c ) ) = P ( G ( c ) ) .intg. 0 .infin. P ( G ( c ) ) G
( c ) . ( 53 ) ##EQU00033##
[0204] In case 1), G.sub.(f)=G. Equation (54) then reduces to
equation (5). The amplitude distribution P.sub.D(D) is determined
based on the amplitude distribution P(G.sub.(1)) using the
following equation:
P D ( D ) = 12 R c 2 t 0 P norm ( G ( 1 ) ) . ( 54. )
##EQU00034##
[0205] For case 2),
G ( - 1 ) = .pi. .eta. R c 2 t 0 2 k B T R h . ##EQU00035##
[0206] The implementation of the constrained regularization
algorithm results in the determination of the amplitude
distribution P(G.sub.(-1)). The distribution P.sub.R(R.sub.h) is
then determined based on P.sub.norm(G.sub.(-1)) according to the
relation:
P R ( R h ) = .pi. .eta. R c 2 t 0 2 k B T P norm ( G ( - 1 ) ) . (
55. ) ##EQU00036##
[0207] Lastly, for case 3),
G ( - ( 1 + a ) / 3 ) = ( 3 K 10 .pi. N a ) ( 1 1 + a ) ( .pi.
.eta. R c 2 t 0 2 k B T ) ( 3 1 + a ) M . ##EQU00037##
[0208] The implementation of the constrained regularization
algorithm results in the determination of the amplitude
distribution P(G.sub.(-(1+a)/3)). The distribution P.sub.M(M) is
then determined by means of P.sub.norm(G.sub.(-(1+a)/3)) according
to the following relation:
P M ( M ) = ( 3 K 10 .pi. N a ) ( 1 1 + a ) ( .pi. .eta. R c 2 t 0
2 k B T ) ( 3 1 + a ) P norm ( G ( - 1 ( 1 + a ) / 3 ) ) . ( 56 )
##EQU00038##
[0209] The manner of selecting the interval [G.sub.(c),min,
G.sub.(c),max] on which the distribution P(G.sub.(c)) is to be
sought is similar to that described above. In particular,
.tau..sub.min and .tau..sub.max are calculated according to
equations (43) and (44). Lastly, the values G.sub.(c),min,
G.sub.(c),maxare determined as follows:
1 ) c = 1 : G min = .tau. max - 2 , G max = .tau. min - 2 . ( 57 )
2 ) c = 1 G min = .tau. max 2 , G max = .tau. min 2 . ( 58 ) 3 ) c
= - 1 / d f = - ( 1 + a ) / 3 G min = .tau. max ( 6 1 + a ) , G max
= .tau. min ( 6 1 + a ) . ( 59 ) ##EQU00039##
Method for Determining the Size Distribution of a Sample
[0210] The method for determining the size distribution of a
mixture of particles will now be described by reference to FIG.
2.
[0211] The method includes a first step 100 of injecting a sample
to be analyzed into the injection section 7 of the experimental
device 3.
[0212] Then, in step 110, after actuating the means 13 for
introducing and circulating an eluent inside the capillary 6, the
sample injected is transported from the injection section 7 to the
detection section 9 of the experimental device 3. The experimental
conditions (nature of the eluent, flow speed of the eluent,
transport distance separating the injection section from the
detection section, temperature, internal ray of the capillary,
etc.) are adapted so that a Taylor dispersion phenomenon will occur
that is detectable in the detection section 9. In the experimental
examples below, precise experimental conditions are indicated.
[0213] In step 120, the sample transported by the eluent passes
through the optical cell of the detection section 9. The sensor 17
then generates an electrical measurement signal characteristic of
the Taylor dispersion occurring in the sample.
[0214] In step 130, the detection signal generated by the sensor 17
is processed by the electronic card 19 so as to deliver a digitized
experimental Taylor signal S(t).
[0215] In step 140, the experimental Taylor signal S(t) is acquired
by the computer 5.
[0216] It is then analyzed (step 200) by running the software 31 in
order to determine a size distribution. The software 31 carries out
the following elementary steps.
[0217] In step 142, a first adapted menu is presented to the user
so that the user may select the parameter according to which the
constrained regularization method is to be carried out. The user
may thus choose either the diffusion coefficient D (case 1, c=1),
the hydrodynamic ray (case 2, c=-1), or the molar mass M (case 3,
c=-1/d.sub.f). The user is also asked to choose the number N for
the discretization of the distribution sought. In the following,
for simplicity, it is assumed that the user chooses the diffusion
coefficient D and that the parameter to be taken into account is
the parameter G.
[0218] In step 144, a second adapted menu is presented to the user
so that the user may select the number and nature of the
constraints to be taken into account in seeking the distribution
that is the solution of equation (5). The proposed constraints to
select are, e.g.: [0219] 1. regularity of distribution; [0220] 2.
the distribution is positive at all points; [0221] 3. the
distribution must result in a predetermined T average plus or minus
a deviation to be specified; [0222] 4. the distribution must result
in a predetermined r average plus or minus a deviation to be
specified.
[0223] In the following, it is assumed that the first constraint is
implemented via a Lagrange multiplier in the cost function, whilst
the other constraints are implemented directly by appropriately
limiting the space of the amplitudes P(G.sub.m) in which an extrema
of the cost function is sought.
[0224] In step 146, the experimental Taylor signal S(t) is broken
down into cumulants. More specifically, the normalized Taylor
signal s(t)=S(t)/S(t.sub.0) is first determined, then its logarithm
ln[s(t)] is calculated. Lastly, a second-degree polynomial of the
variable (t-t.sub.0).sup.2 is adjusted to the function ln[s(t)].
The first- and second-order cumulants .GAMMA..sub.1 and
.GAMMA..sub.2 are then determined. .GAMMA..sub.1 allows, in
particular, for a measurement of the .GAMMA. average of the
parameter G. Additionally, the T average of the Taylor signal is
measured.
[0225] In step 148, the limits G.sub.min and G.sub.max of the
interval of the values of the parameter G on which the distribution
is sought are calculated based on the equivalent log-normal
distribution determined based on the values of the first- and
second-order cumulants .GAMMA..sub.1 and .GAMMA..sub.2 obtained in
step 148, using equations (32) and (33) followed by (38) and
(39).
[0226] In step 150, the cost function H.sub..alpha. is developed
from the first constraint selected by the user in step 144. The
constraint term associated with the constraint selected is red in
the memory of the computer 5.
[0227] In step 152, the discretized expression of the cost function
H.sub..alpha. is obtained by subdividing the interval G.sub.min and
G.sub.max determined in step 154 into N sub-intervals.
[0228] In step 154, for each value of the Lagrange coefficient
.alpha. in a group of test values,
[0229] the minimum of the cost function H.sub..alpha. is
determined. To take into account strict constraints according to
which the amplitudes are positive and must result in predetermined
T and .GAMMA. averages with a predetermined deviation, the minimum
of the cost function is sought exclusively on the appropriate
subspace of the amplitudes P(G.sub.m) that satisfy these strict
constraints.
[0230] In step 156, the statistical error .nu. is determined, and
the optimal value .alpha..sub.0 of the Lagrange coefficient .alpha.
is determined by selecting the value of the Lagrange coefficient
.alpha., which, in step 156, resulted in the nearest distance term
.chi..sub.2 by values lower than this statistical error .nu..
[0231] In step 158, the group of distributions P(G) sought is the
group of those that minimise the cost function H.sub..alpha. for
the optimal value .alpha..sub.0 of the Lagrange coefficient
[0232] determined in step 156.
[0233] In step 160, the value related to the size of the particles
of the mixture is calculated based on the distribution P(G)
obtained in step 158.
[0234] Lastly, in step 162, for an adapted transformation, the
distributions according to the hydrodynamic ray or molar mass are
calculated based on the distribution P(G) obtained in step 158.
[0235] If applicable, the various distributions calculated are
displayed on the screen of the computer 5. The software 31 includes
`tools` allowing the user to carry out the desired calculations on
the calculated distributions.
[0236] The software 31 thus includes means suited for the execution
of each of the steps of the analysis of the experimental Taylor
signal.
[0237] In one variant, the limits G.sub.min and G.sub.max of the
interval on which the distribution P(G) is sought are calculated
empirically. This consists of determining the normalized Taylor
signal s(t)=S(t)/S(t.sub.0), obtaining its logarithm, and then
calculating the derivative |.differential.ln s/.differential.x|.
The limits of the interval of interest of the parameter G are
finally deduced using equations (43) and (44), followed by
equations (46) and (47).
[0238] In yet another variant, independent of the preceding
variant, the T average of the parameter G, G.sub.T, is calculated
by integrating the experimental Taylor signal S(t) using equation
(16), and the .GAMMA. average of the parameter G, G.sub..GAMMA., is
calculated based on the determination of the first-order cumulant
resulting from the breakdown of the experimental Taylor signal into
cumulants. The limits G.sub.min, and G.sub.max of the interval on
which the size distribution is to be sought are calculated based on
the T and .GAMMA. of the parameter G according to equations (30),
(31), and (38), (39). This variant of the method also allows for a
constraint term based on one or the other of these averages to be
integrated into the cost function.
Constrained Regularization Analysis on a Series of Experimental
Taylor Signals of a Single Sample
[0239] Adjustment by constrained regularization may be
advantageously implemented using several experimental Taylor
signals obtained by repeating identical experiments on a group of
samples of a single mixture.
[0240] Although each repetition may be analyzed independently, and
the amplitude distributions obtained may be averaged, it has been
found to be more robust to accumulate the various individual Taylor
signals into a single global Taylor signal including a number of
experimental points equal to the sum of the experimental points of
each individual Taylor signal. Secondly, the amplitude distribution
is sought on the global Taylor signal by applying the constrained
regularization algorithm.
[0241] This results in an amplitude distribution P(G) that most
closely observes the constraint imposed, e.g., the distribution
P(G) is more regular. This also allows the uncertainties and
imprecisions affecting the acquisition of the individual Taylor
signals to be taken into account.
[0242] During this operation, if the reference time t.sub.0 is not
strictly identical from one experimental Taylor signal to another,
the time coordinates are translated such that all experimental
Taylor signals have exactly the same reference time t.sub.0.
[0243] The correction of the baseline, followed by the
normalization of each experimental Taylor signal, may also be
necessary before processing.
[0244] The software 31 includes a menu allowing users to process
several experimental Taylor signals before analysing the global
Taylor signal thus obtained.
Advantages
[0245] The method just discussed allows the size distribution of a
mixture of species, as well as the concentrations of these species
in the mixture, to be obtained automatically and in real time, no
matter what the polydispersity of the sample analyzed is, i.e., the
number of species included in this sample and the respective
concentrations thereof.
[0246] The fields of application of the device and method described
above include the size characterization of polymers, colloids,
latex nanomaterials, emulsions, liposomes, vesicles, and molecules
or biomolecules in general. One important field of application is
the study of the stability/degradation/aggregation of proteins for
the pharmaceuticals industry.
[0247] The advantages of the characterization of a sample by means
of the Taylor dispersion phenomenon are known to persons skilled in
the art: Low volume of the sample to be injected into the
capillary, no need to calibrate the experimental device, use of an
extremely simple experimental device, technique that is
particularly well adapted to size measurements of particles smaller
than a few nanometres, a signal that is generally sensitive to mass
concentration, etc.
EXPERIMENTAL EXAMPLES
Experimental Conditions
[0248] Virgin silicon capillary: R.sub.c=50 .mu.m with a distance
between the injection and detection sections of 30 cm.
[0249] Temperature: T=293.degree. K
[0250] Eluent: Sodium borate buffer 80 mM, pH 9.2.
[0251] Viscosity of the eluent: .eta.=8.9 10.sup.-4 Pas.
[0252] Sample: Polystyrene sulphonate (PSS) 0.5 g/l.
[0253] Injection: 0.3 psi (20 mbar), 9 s, i.e., an injected volume
of 8 nl (total capillary volume 589 nl).
[0254] UV detection at a wavelength of 200 nm.
Characteristics of the Polymer Standards Injected
[0255] 1 ) PSS 1290 M w = 1290 g / mol M p = 1094 g / mol M w / M n
< 1.20 2 ) PSS 1590 M w = 5190 g / mol M p = 5280 g / mol M w /
M n < 1.20 3 ) PSS 29000 M w = 29000 g / mol M p = 29500 g / mol
M w / M n < 1.20 4 ) PSS 148000 M w = 145000 g / mol M p =
148500 g / mol M w / M n < 1.20 5 ) PSS 333000 M w = 333000 g /
mol M p = 338000 g / mol M w / M n < 1.20 ##EQU00040##
where M.sub.w is the average molar mass by weight, M.sub.p the
molar mass at the summit of the chromatographic peak, and
M.sub.w/M.sub.n is the polydispersity index. The average molar
masses and the characteristics of the distribution were given by
the supplier, who determines them by steric exclusion
chromatography with calibration using polymer standards of the same
chemical nature (PSS).
Experimental Taylor Signals of the Polymers Studied
[0256] FIG. 3 shows an experimental Taylor signal obtained for an
equimassic mixture of PSS 1290 and PSS 29000. In fact, three
experimental Taylor signals are aggregated here.
[0257] Furthermore, only the left part of the experimental Taylor
signal is shown. In fact, generally, taylorgrams are symmetrical.
However, in the case of the phenomenon of adsorption to the
capillary surface, the right part of the signal, corresponding to
the times following the time t.sub.0, may not be exactly
symmetrical on the left part of the signal, corresponding to the
time preceding the time t.sub.0. In this case, it is preferable to
focus the
[0258] processing of the data on the left part of the experimental
Taylor signal. Advantageously, the method described above only
takes into account the left part of the signal in order to limit
the influence of these possible parasitic phenomena.
[0259] FIG. 4 shows the superimposition of the distribution
P.sub.R(R.sub.h) obtained by the aforementioned method (software
31) compared to the distribution given by the supplier of the
polymers and obtained by a chromatographic method (SEC). FIG. 4
shows good consistency between the hydrodynamic ray distribution
given by the supplier and the distribution obtained by running the
software 31.
[0260] FIG. 5A shows the adjustment between the normalized
experimental Taylor signal S(t) (Data) and the normalized
reconstructed signal S'(t) (Fit), and FIG. 5B shows the
[0261] adjustment between the logarithm of the normalized
experimental Taylor signal S(t)(Data) and the cumulant development
(Fit).
Comparison of the Results
[0262] Table 1 shows the various averages of the diffusion
coefficient D obtained: Directly
[0263] by breakdown into cumulants (.GAMMA. average, column 2) or
by integrating the taylorgram (7 average, column 3), and, on the
other hand, by running the software 31 (.GAMMA. average, column 4
and Taverage, column 5). It should be noted that, in this example,
the averages measured directly on the experimental Taylor signal
are not used to constrain the deconvolution of the signal, and only
a loose regularity constraint and a strict positivity constraint
were used.
TABLE-US-00001 TABLE 1 1 2 3 4 5 Sample: D .sub..GAMMA. .sup.a D
.sub.T .sup.b D .sub..GAMMA. .sup.c D .sub.T .sup.c 1290 2.534E-06
2.457E-06 2.654E-06 2.595E-06 5190 1.197E-06 1.083E-06 1.197E-06
1.192E-06 29000 4.754E-07 4.222E-07 5.152E-07 5.135E-07 145000
2.014E-07 1.899E-07 2.218E-07 2.201E-07 333000 1.209E-07 1.157E-07
1.413E-07 1.119E-07 1290 + 29000 1.065E-06 8.407E-07 1.900E-06
8.533E-07 1290 + 5190 2.137E-06 1.555E-06 2.082E-06 1.812E-06 5190
+ 29000 7.669E-07 6.509E-07 9.237E-07 6.921E-07 .sup.a based on the
breakdown into cumulants (equation 25) .sup.b based on the
integration of the taylorgram (equation 17) .sup.c based on the
distribution given by the software 31
[0264] Overall, the results show great coherence, and the software
31 results in a solution in which the T and .GAMMA. averages
(columns 4 and 5) are close to the experimental values (columns 2
and 3).
[0265] Table 2 shows the values of .tau..sub.min and .tau..sub.max
determined based on the various proposed approaches, i.e., the
imperical approach (columns 4 and 5), breakdown into cumulants
(columns 6 and 7) based on the cumulants .GAMMA..sub.1 and
.GAMMA..sub.2 (columns 2 and 3), and the approach using the T and
.GAMMA. averages of the diffusion coefficient (columns 8 and 9
based on the first-order cumulant and the integration of the
taylorgram).
TABLE-US-00002 TABLE 2 2 3 4.sup.a 5.sup.a 6.sup.b 7.sup.b 8.sup.c
9.sup.c 1 .GAMMA..sub.1 .GAMMA..sub.2 .tau..sub.min/.alpha..sub.min
.tau..sub.max/.alpha..sub.max .tau..sub.min/.alpha..sub.min
.tau..sub.max/.alpha..sub.max .tau..sub.min/.alpha..sub.min
.tau..sub.max/.alpha..sub.max Sample: s.sup.-2 s.sup.-4 s.sup.a
s.sup.a s.sup.b s.sup.b en s en s 1290 6.339E-02 1.231E-05 3.891
4.280 3.679 4.301 3.277 4.914 5190 3.004E-02 5.638E-06 5.554 5.985
5.176 6.471 4.137 8.603 29000 1.189E-02 6.307E-06 7.595 9.052 6.975
12.593 6.408 14.203 145000 5.038E-03 1.651E-06 9.804 13.555 10.195
20.736 10.856 19.016 333000 3.024E-03 5.378E-07 11.279 30.264
13.344 26.235 14.497 23.480 1290 + 29000 2.664E-02 8.047E-05 4.704
8.860 4.067 10.276 3.781 11.622 1290 + 5190 5.210E-02 2.518E-04
4.453 7.740 3.005 6.978 2.540 9.343 5190 + 29000 1.918E-02
2.571E-05 6.625 9.891 5.171 10.784 4.777 12.172 .sup.aempirical
approach based on equations (43) and (44) .sup.bbased on the
cumulant breakdown (equations (32-33), (38-39), and (46-47))
.sup.cbased on the T and .GAMMA. averages of the diffusion
coefficient (equations (17), (25), (30-31), (38-39), and
(46-47)).
[0266] The orders of magnitude of .tau..sub.min and .tau..sub.max
are highly consistent no matter what method is considered.
[0267] Table 3 compares the average hydrodynamic ray values
obtained by breakdown into cumulants (.GAMMA. average, column 2),
and by running the software 31 by determining the distribution P(G)
followed by the .GAMMA. T integration (columns 3 and 6), by the
reference method by steric exclusion chromatography (columns 4 and
7) following the average in question, by direct integration of the
taylorgram on the entire signal (column 5). For the same average
(columns 2-4, on the one hand, and column 5-7, on the other hand),
the results are homogeneous for all samples considered.
[0268] This shows high consistency for all samples for each group
of averages in question.
TABLE-US-00003 TABLE 3 1 2 3 4 5 6 7 Sample: kT 6 .pi..eta. [ D
.GAMMA. ] cumulant ##EQU00041## kT 6 .pi..eta. [ D .GAMMA. ] log
iciel 31 ##EQU00042## kT 6 .pi..eta. [ D .GAMMA. ] SEC ##EQU00043##
kT 6 .pi..eta. [ D T ] Taylor ##EQU00044## kT 6 .pi..eta. [ D T ]
Logiciel 31 ##EQU00045## kT 6 .pi..eta. [ D T ] SEC ##EQU00046##
1290 0.968 0.924 0.913.sup.a 0.998.sup.b 0.945.sup.c 1.000.sup.d
5190 2.049 2.048 1.880 2.265 2.057 1.989 29000 5.159 4.761 5.076
5.810 4.776 5.278 145000 12.178 11.057 12.200 12.917 11.144 13.133
333000 20.286 17.359 17.297 21.190 21.917 20.915 1290 + 29000 2.303
1.291 1.319 2.917 2.874 3.449 1290 + 5190 1.148 1.178 1.046 1.578
1.354 1.470 5190 + 29000 3.198 2.655 2.473 3.768 3.544 3.638
.sup.aobtained based on the distribution obtained by SEC.
.sup.bobtained by integrating the taylorgram (based on the variance
of the taylorgram). .sup.cobtained by integrating the weight
distribution of the diffusion coefficients obtained by the software
31 (minimisation on D). .sup.dobtained based on the weight
distribution of the hydrodynamic rays originating from the SEC.
Determination of the Time at the Peak
[0269] Experimentally, the time at the peak to of the experimental
Taylor signal S(t) is not known with precision due to the
measurement noise.
[0270] The time at peak t.sub.0 affects both a cumulant analysis
and the determination of the size distribution obtained by
constrained regularization.
[0271] Additionally, the cumulant method is based on a limited
development for (t-t.sub.0).fwdarw.0. From an experimental
standpoint, it is necessary to choose the range of time t for the
analysis wisely: If one limits oneself to a very small interval,
the result will be substantially affected by measurement noise. If,
on the other hand, too wide an interval is considered, the
contribution of the higher-order (t-t.sub.0) terms, which are
ignored in the cumulant method, will be significant.
[0272] A step for determining a peak time t.sub.0, as well as an
optimal range of times suitable to a cumulant analysis, is shown
below.
[0273] In a first sub-step, a first estimate t.sub.0,guess of the
peak time is obtained, e.g., by considering the time for which S(t)
is at its maximum or by adjusting the peak of S(t) by means of a
parabolic or Gaussian function.
[0274] In a second sub-step, a list of N peak times to be tested
t.sub.0,i is established, with the natural integer i varying
between 1 and N, where the times t.sub.0,i are around t.sub.0,guess
and regularly spaced by a constant time increment, with:
t.sub.0,1<t.sub.0,2<. . . t.sub.0,N;
t.sub.0,1+1=t.sub.0,i+dt; and
t.sub.0,1=t.sub.0,guess-.DELTA.t,
t.sub.0,N=t.sub.0,guess+.DELTA.t;
[0275] where
[0276] df is the time increment between two consecutive tested peak
times; and
[0277] .DELTA.t is a time interval typically on the order of
t.sub.0,guess/50.
[0278] In a third sub-step, for each of the peak times to be tested
t.sub.0,i, a series of cumulant analyses is carried out taking into
account various ranges of time of differing lengths.
[0279] The time range is, e.g., at a cutoff level of the signal
S(t). For example, for a cutoff level of 0.1, the time range t is
considered such that S(t)>0.1.times.S(t.sub.0,i). The value of
the first and second cumulant resulting from the adjustment on each
range of time is noted.
[0280] In a fourth sub-step, the optimal peak time t.sub.0 is
determined as being between the peak times for which the first
cumulant .GAMMA..sub.1 diverges towards positive values when the
cutoff level increases, and those for which the first cumulant
.GAMMA..sub.1 diverges towards negative values when the cutoff
level increases.
[0281] Tracing on a graph, for each peak time to be tested
t.sub.0,i, the curve of the first cumulant .GAMMA..sub.1 as a
function of the cutoff level, the optimal peak time t.sub.0 is that
for which the curve is located between upward concave curves and
downward concave curves. This curve has a smaller variation than
the others.
[0282] The choice of optimal peak time is made by visual analysis
of the aforementioned graphics or automatically, e.g., based on the
sign of the second numerical derivative of the first cumulant
.GAMMA..sub.1 as a function of the cutoff.
[0283] Alternatively or optimally, it is possible to do the same
with the second cumulant .GAMMA..sub.2 and/or the square of the
ratio of the second cumulant to the first cumulant. By doing it
simultaneously for the first cumulant, the second cumulant, and the
square of the ratio of the second cumulant to the first cumulant,
the choice of peak time may be made more reliable.
[0284] In a fifth step, the optimal cutoff value is determined as
the one that is the highest before the data show a significant
deviation relative to their general tendancy due to the influence
of measurement noise for very high cutoff levels.
* * * * *